A123511 Arises in the normal ordering of functions of a*(a+)*a, where a and a+ are the boson annihilation and creation operators, respectively.

1, 8, 70, 680, 7315, 86576, 1119468, 15710640, 237885285, 3865865080, 67113398066, 1239550196248, 24267176759735, 501941612835040, 10936819334789720, 250370971426742496, 6007479214999260873

Offset: 0

Author

Karol A. Penson, Oct 02 2006

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..440

Crossrefs

Cf. A002720, A052852, A123510, A123512.

Programs

Mathematica

max = 16; s = (1/(1 - x)^4)*Exp[x/(1 - x)]*LaguerreL[3, -x/(1 - x)] + O[x]^(max + 1); CoefficientList[s, x]*Range[0, max]! (* Jean-François Alcover, May 23 2016 *)

Formula

E.g.f.: (1/(1-x)^4)*exp(x/(1-x))*LaguerreL(3,-x/(1-x)).

From Vaclav Kotesovec, Nov 13 2017: (Start)

Recurrence: n*a(n) = 2*n*(n+3)*a(n-1) - (n-1)*(n+2)*(n+3)*a(n-2).

a(n) ~ exp(2*sqrt(n)-n-1/2) * n^(n + 13/4) / (3*2^(3/2)) * (1 + 31/(48*sqrt(n))).

(End)

Extensions

a(0)=1 prepended by G. C. Greubel, Oct 31 2017